Optimal. Leaf size=665 \[ \frac{7 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt [3]{d} \sqrt{c+d x^3}}{432 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{216 \sqrt{3} c^{17/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{648 c^{17/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{648 c^{17/6}} \]
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Rubi [A] time = 0.828396, antiderivative size = 665, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {472, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{7 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt [3]{d} \sqrt{c+d x^3}}{432 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{216 \sqrt{3} c^{17/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{648 c^{17/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{648 c^{17/6}} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
Rule 584
Rule 303
Rule 218
Rule 1877
Rule 486
Rule 444
Rule 63
Rule 206
Rule 2138
Rule 2145
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx &=\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}+\frac{\int \frac{28 c d+\frac{5 d^2 x^3}{2}}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{216 c^2 d}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\int \frac{x \left (-160 c^2 d^2+14 c d^3 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{1728 c^4 d}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\int \left (-\frac{14 c d^2 x}{\sqrt{c+d x^3}}-\frac{48 c^2 d^2 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{1728 c^4 d}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}+\frac{(7 d) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{864 c^3}+\frac{d \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{36 c^2}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\int \frac{2 \sqrt [3]{c} d^{2/3}-2 d x-\frac{d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right ) \sqrt{c+d x^3}} \, dx}{432 c^3}+\frac{\left (7 d^{2/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{864 c^3}+\frac{d^{2/3} \int \frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{c+d x^3}} \, dx}{432 c^{8/3}}+\frac{\left (7 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{2/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{432 c^{8/3}}-\frac{d^{4/3} \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{144 c^{7/3}}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt [3]{d} \sqrt{c+d x^3}}{432 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{7 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{1}{9-c x^2} \, dx,x,\frac{\left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt{c+d x^3}}\right )}{216 c^{7/3}}-\frac{d^{4/3} \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{432 c^{7/3}}+\frac{d^{7/3} \operatorname{Subst}\left (\int \frac{1}{-\frac{2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{c+d x^3}}\right )}{108 c^{10/3}}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt [3]{d} \sqrt{c+d x^3}}{432 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{216 \sqrt{3} c^{17/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{648 c^{17/6}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{7 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{216 c^{7/3}}\\ &=-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt [3]{d} \sqrt{c+d x^3}}{432 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left (8 c-d x^3\right )}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{216 \sqrt{3} c^{17/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{648 c^{17/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{648 c^{17/6}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{7 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}\\ \end{align*}
Mathematica [C] time = 0.122201, size = 180, normalized size = 0.27 \[ \frac{7 d^2 x^6 \left (d x^3-8 c\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+200 c d x^3 \left (8 c-d x^3\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-80 c \left (54 c^2+47 c d x^3-7 d^2 x^6\right )}{34560 c^4 \sqrt{c+d x^3} \left (8 c x-d x^4\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.014, size = 1761, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{d^{3} x^{11} - 15 \, c d^{2} x^{8} + 48 \, c^{2} d x^{5} + 64 \, c^{3} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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